Packing Hamilton Cycles Online
arXiv:1608.04976v1 [math.CO] 17 Aug 2016
Joseph Briggs∗, Alan Frieze†, Michael Krivelevich‡, Po-Shen Loh§, Benny Sudakov¶
Abstract It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge disjoint Hamilton cycles, [4], [13], [9]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ≥ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to color the edges online with σ colors so that at time τ2σ , each color class is Hamiltonian.
1
Introduction
The celebrated random graph process, introduced by Erd˝os and Renyi [5] in the 1960’s, begins with an empty graph on n vertices, and at every step t = 1, . . . , n2 adds to the current graph a single new edge chosen uniformly at random out of all missing edges. Taking a snapshot of the random graph process after m steps produces the distribution Gn,m . An equivalent “static” way of defining Gn,m would be: choose m edges uniformly at random out of all n2 possible ones. One advantage in studying the random graph process, rather than the static model, is that it allows for a higher resolution analysis of the appearance of monotone graph properties (a graph property is monotone if it is closed under edge addition). A Hamilton cycle of a graph is a simple cycle that passes through every vertex of the graph, and a graph containing a Hamilton cycle is called Hamiltonian. Hamiltonicity is one of the most fundamental notions in graph theory, and has been intensively studied in various contexts, including random graphs. The earlier results on Hamiltonicity of random graphs were obtained by P´osa [15], and Korshunov [10]. Improving on these results, Bollob´as [3], and Koml´os and Szemer´edi [11] proved that if m0 = 21 n log n + 12 n log log n + nω(n), for any ω(n) tending to infinity together with n, then Gn,m0 is Hamiltonian w.h.p. One obvious necessary condition for the graph to be Hamiltonian is for the minimum degree to be at least 2, and the above result indicates that the events of being Hamiltonian and of having all degrees at least two are indeed bundled together closely. Bollob´ as [3], and independently, Ajtai, Koml´os, and Szemer´edi [1], further strengthened this by proving that w.h.p. the random graph process becomes Hamiltonian when the last vertex ∗
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA.
[email protected]. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA.
[email protected]. Research supported in part by NSF grant DMS1362785. ‡ School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel.
[email protected]. Research supported in part by a USA-Israel BSF Grant and by a grant from Israel Science Foundation. § Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.
[email protected]. Research supported in part by NSF grant DMS-1455125. ¶ Department of Mathematics, ETH, 8092 Zurich, Switzerland.
[email protected]. Research supported in part by SNSF grant 200021-149111. †
1
of degree one disappears. A more general property Hσ of having σ edge disjoint Hamilton cycles was studied by Bollob´ as and Frieze [4]. They showed that if σ = O(1) then w.h.p. the random graph process satisfies Hσ when the minimum degree becomes 2σ. It took quite a while, but this result was extended to the more difficult case of growing σ in the Gn,m context by Knox, K¨ uhn and Osthus [9] and Krivelevich and Samotij [13]. Recently, quite a lot of attention and research effort has been devoted to controlled random graph processes. In processes of this type, an input graph or a graph process is usually generated fully randomly, but then an algorithm has access to this random input and can manipulate it in some well defined way (say, by dropping some of the input edges, or by coloring them), aiming to achieve some preset goal. There is usually the so-called online version where the algorithm must decide on its course of action based only on the history of the process so far and without assuming any familiarity with future random edges. For example, in the so-called Achlioptas process the random edges arrive in batches of size k. An online algorithm chooses one of them and puts it into the graph. By doing this one can attempt to accelerate or to delay the appearance of some property. Hamiltonicity in Achlioptas processes was studied in [12]. Another online result on Hamiltonicity was proved in [14]. There, it was shown that one can orient the edges of the random graph process so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time when the underlying graph has minimum degree two. Here we consider a Ramsey-type version of controlled random processes. In this version, the incoming random edge, when it is exposed, is irrevocably colored by an algorithm in one of r colors, for a fixed r ≥ 2. The goal of the algorithm is to achieve or to maintain a certain monotone graph property in all of the colors. For example, in [2] the authors considered the problem of creating a linear size (so-called giant) component in every color. The above mentioned result of Bollob´as and Frieze [4] gives rise to the following natural question. Can one typically construct σ edge disjoint Hamilton cycles in an online fashion by the time the minimum degree becomes 2σ? We answer this question affirmatively in the case σ = O(1). Theorem 1.1. For a fixed integer σ ≥ 2, let τ2σ denote the hitting time for the random graph process Gi , i = 1, 2, . . . to have minimum degree 2σ. Then w.h.p. we can color the edges of Gi , i = 1, 2, .. online with σ colors so that Gτ2σ contains σ Hamilton cycles C1 , C2 , . . . , Cσ , where the edges of cycle Cj all have color j.
2
Description of the coloring procedure
We describe our coloring procedure in terms of q = 2σ colors we aim to color the edges so that each vertex has degree at least one in each color. Think of colors 1 and 1 + σ being light red and dark red, say, and then that each vertex is incident with at least two red edges. This may appear cumbersome, but it does make some of the description of the analysis a little easier. In the broadest terms, we construct two sets of edges E + and E ∗ . Let Γ∗c be the subgraph of Gτ2σ induced by the edges of color c in E ∗ . We ensure that w.h.p. this has minimum degree at least one for all c. We then show that w.h.p. after merging colors c and c + σ for c ∈ [σ] the ∗ ∗ subgraph Γ∗∗ c = Γc ∪ Γc+σ has sufficient expansion properties so that standard arguments using P´osa rotations can be applied. For every color c, the edges of Ec∗ are used to help create a good expander, and produce a backbone for rotations. And the edges in Ec+ are used to close cycles in this argument. Notation. “At time t” is taken to mean ”when t edges have been revealed”. (t)
Notation. Let N (t) (v) denote the set of neighbors of v in Gt and let dv = |N (t) (v)|. 2
For color c ∈ [q], write dc = dc,t , Nc = Nc,t for the degrees and neighborhoods of vertices and sets in Γc . Definition 2.1. Let F ull denote the set of vertices with degree at least time t := n log n ,
log n 1000q
in every color at
where is some sufficiently small constant depending only on the constant q. The actual value of needed will depend on certain estimates below being valid, in particular equation (14). A vertex is F ull if is lies in F ull. Similarly, let F ull0 ⊆ F ull denote the set of vertices with degree at least log n 1 1000q in every color at time 2 n log n.
2.1
Coloring Algorithm COL
We now describe our algorithm for coloring edges as we see them. At any time t, vertex v has a list (t) (t) Cv := {c ∈ [q] : dc (v) = 0} of colors currently not present among edges incident to v; “the colors (t) that v needs”. A vertex is needy at time t if Cv 6= ∅. If the next edge to color contains a needy vertex then we try to reduce the need of this vertex. Otherwise, we make choices to guarantee expansion in E ∗ , needed to generate many endpoints in the rotation pahse, and to provide edges for E + , which are used to close cycles, if needed. FOR t = 1, 2, . . . , τq DO BEGIN Step 1 Let et = uv. (t)
(t)
(t)
(t)
(t)
(t)
Step 2 If Cv ∪ Cu = ∅, t > t , and precisely one of {u, v} (WLOG u) is F ull, then give uv the color c that minimises dc (v) (breaking ties arbitrarily). Add uv to Ec∗ . Step 3 If Cv ∪ Cu = ∅, t > t and both u, v ∈ F ull, give uv a color c uniformly at random from [q]. Then add this edge to Ec+ or Ec∗ , each with probability 1/2. Step 4 If Cv ∪ Cu = ∅ but t ≤ t or both u, v ∈ / F ull, then color uv with color c chosen uniformly at random from [q]. Add uv to Ec∗ . Step 5 Otherwise, color uv with color c chosen uniformly at random from (t) (t) Cu ∪ Cv . Add uv to Ec∗ . END (t)
(t)
Remark 2.2. Observe that if Cv 6= ∅ then uv is colored with c ∈ Cv with probability at least 1 q . This holds regardless of the previous history of the algorithm.
3 Let
Structural properties log n + (q − 1) log log n − ω p= and m = n
where ω = ω(n) → ∞, ω = o(log log n). 3
n p 2
We will use the following well-known properties relating Gn,p and Gn,m , see for example [7], Chapter 1. Let P be a graph property. It is monotone increasing if adding an edge preserves it, and is monotone decreasing if deleting an edge preserves it. We have: P(Gn,m ∈ P) ≤ 10m1/2 P(Gn,p ∈ P).
(1)
P(Gn,m ∈ P) ≤ 3P(Gn,p ∈ P), if P is monotone.
(2)
n A vertex v ∈ [n] is small if its degree d(v) in Gn,m satisfies d(v) < log 100q . It is large otherwise. The set of small vertices is denoted by SM ALL and the set of large vertices is denoted by LARGE.
Definition 3.1. A subgraph H of Gn,m with a subset S(H) ⊂ V (H) is called a small structure if |E(H)| + |S(H)| − |V (H)| ≥ 1. We say that Gn,m contains H if there is an injective homomorphism φ : H ,→ Gn,m such that φ(S(H)) ⊆ SM ALL. The important examples of H include: • A single edge between 2 small vertices. • A path of length at most five between two small vertices. • A copy of C3 or C4 with at least one small vertex. • Two distinct triangles sharing at least one vertex. Lemma 3.2. For any fixed small structure H of constant size, P(Gn,m contains H) = o(n−1/5 ). Proof. We will prove that P(Gn,p contains H) = o(n−3/4 ). This along with (1) implies the lemma. Let h = |V (H)|, f = |E(H)|, s = |S(H)| so that f + s ≥ h + 1. Then: log n s 100q n X n − h i P (Gn,p contains H) ≤ h!pf p (1 − p)n−h−i h i i=0
. nh
≤ nh
log n n
f
log n n
f
log n s i 100q X (e + o(1)) log n e− log n−(q−1) log log n+ω+o(1) i i=0
log n
(300q) 100q n(log n)q−1−o(1)
!s
= o(nh−f −s+1/4 ) = o(n−3/4 ). (We used the notation A . B in place of A ≤ (1 + o(1))B.) In the calculation above, in the first line we placed the vertices of H and decided about the identity of s vertices falling into SM ALL, then required that all f edges of H are present in Gn,p , and finally required that for each of the s n vertices in SM ALL, their degree outside the copy of H is at most log 100q .
4
h i n Lemma 3.3. W.h.p., for every k ∈ q − 1, log 100q , there are less than νk = degree k in Gn,m .
e2ω (log n)k−q+1 (k−1)!
Remark 3.4. νk is increasing in k for this range, and for the largest k = n
log(100eq) 100q
log n 100q
vertices of
we have νk .
.
Proof. Fix k and then we have P(Gn,p has at least νk vertices of degree at most k) !νk X k n n − νk ` ≤ p (1 − p)n−νk −` νk ` `=0 νk n n − νk k = (1 + o(1)) p (1 − p)n−νk −k νk k !νk ne nk log n + (q − 1) log log n − ω k − log n−(q−1) log log n+ω+o(1) ≤ × e νk k! n !νk e−ω+O(1) (log n + q log log n)k ≤ (log n)q−1 k!νk !νk e−ω+O(1) q log log n k = 1+ k log n !νk kq/ log n (log n) ≤ e−ω+O(1) . k The function f (k) = (specifically f (q − 1)
(log n)kq/ log n k = eO(1) > f
and so f is maximised at the extreme values of k is log-convex, log n 100q = o(1)). Hence, log n
P(∃k : Gn,p has at least νk vertices of degree k) ≤
100q X
e−ωνk /2 = o(1).
k=q−1
Applying (2) we see that P(∃k : Gn,m has at least νk vertices of degree k) = o(1), which is stronger than required. Lemma 3.5. With probability 1 − o(n−10 ), Gn,m has no vertices of degree ≥ 20 log n. Proof. We will prove that w.h.p. Gn,p has the stated property. We can then obtain the lemma by applying (2). n−1 P(∃v : d(v) ≥ 20 log n) ≤ n p20 log n 20 log n en 2 log n 20 log n ≤n 20 log n n e 20 log n ≤n 10 = o(n−10 ).
5
4
Analysis of COL
Let Γ = Gm and let d(v) denote the degree of v ∈ [n] in Γ. Let ( 0 d(v) ≥ q. θv = 1 d(v) = q − 1. (m)
Lemma 4.1. Suppose we run COL as described above. Then w.h.p. |Cv
| = θv for all v ∈ [n].
In words, Lemma 4.1 guarantees that the algorithm COL typically performs so that at time m, each vertex of degree at least q has all colors present at the incident edges, while each vertex of degree q − 1 has exactly one color missing. (It is easy to argue that w.h.p. δ(Gm ) = q − 1.) Proof. Fix v and suppose v has k neighbours in LARGE, via edges {fi = vui }ki=1 . Then in general d(v) − 1 ≤ k ≤ d(v) as small vertices do not share a path of length two. Also, when v is small, k = d(v). Write t(e) for the time t ∈ [1, m] at which an edge e appears in the random graph process, i.e. t(ei ) = i. Then let ti = t(fi ) and assume that ti < ti+1 for i > 0. Every time an fi (m) appears while ui needs no additional colors, v gets a color it needs. So for v to have |Cv | > θv at the end of the process, this must happen at most q − 2 − θv times, so there is certainly some set of k − q + 2 + θv edges which fail. We omit i = 1 in this consideration since v will always get a color it needs at time t1 . Let TS denote {ti : i ∈ [2, k] \ S}, and let L denote the occurrence of the bound of 20 log n on the degree of v and its neighbors (see Lemma 3.5). Note that P(L) = 1 − o(n−10 ). Taking a union bound over all S, ^ X X 1 (m) P P(|Cv | > θv | TS , L) ≤ Ai TS , L m S⊂[2,k] ti :i∈[k]\S |S|=q−2−θv
X
=
X
S⊂[2,k] ti :i∈[2,k]\S |S|=q−2−θv
since there are ti2 − 1 choices for i1 . Here
k−q+2+θv
t i2 − 1 m k−q+2+θv
i∈[2,k]\S
P
^ i∈[2,k]\S
Ai TS , L ,
(3)
(ti ) Ai := Cui 6= ∅ ,
and {1 = i1 < i2 < · · · < is } are the elements of [k] \ S in increasing order for a fixed S, where s := k − |S| = k − q + 2 + θv . Next let Yi = {edges of ui that appeared before ti excluding edges contained in N (m) (v)}, dr = d(ur ) and Zr := |Yr | for r = 2, 3, . . . , s, DS = {di : i ∈ [2, k] \ S} . Then, P (Ai2 ∧ · · · ∧ Ais | TS , DS , L) X = P(Ais | Ai2 , . . . , Ais−1 , Zs = zs , TS , DS , L) P(Ai2 , . . . , Ais−1 , Zs = zs | TS , DS , L) {z } | zs ≤P(Bin(zs ,q −1 )≤q−1) by Remark 2.2
6
≤
X
g(zs )
X
P(Ais−1 | Ai2 , . . . , Ais−2 , Zs−1 = zs−1 , Zs = zs , TS , DS , L)
zs−1
zs
× P(Ai2 , . . . , Ais−2 , Zs−1 = zs−1 , Zs = zs | TS , DS , L) ≤
X
g(zs )g(zs−1 )P(Ai2 , . . . , Ais−2 , Zs−1 = zs−1 , Zs = zs | TS , DS , L)
zs ,zs−1
≤
X
g(zs ) · · · g(z2 )P(Zr = zr , r = 2, . . . , s | TS , DS , L) (by induction)
(4)
zs ,...,z2
Here g(z) := P(Bin(z, q −1 ) ≤ q − 1) for any z ≥ 0. Claim 4.2.
˜ P(Zr = zr , r = 2, 3, . . . , s | TS , DS , L) ≤ 1 + O(n
−1
s Y )
tr zr
r=2
m−tr dr −zr m dr
,
˜ hides polylog factors. where O Proof
Fix
log n 100q
≤ d1 , d2 , . . . , ds = O(log n) and t2 , t3 , . . . , ts . Then, for every 1 < r ≤ s, tr m−tr zr dr −zr m−d2 −···−dr−1 −s dr tr m−tr d −z z r r ˜ −1 ) r . 1 + O(n m dr
P(Zr = zr | Zr−1 = zr−1 , . . . , Z2 = z2 , TS , DS , L) ≤ ≤
(5)
Explanation for (5): The the first binomial coefficient in the numerator in (5) bounds the number of choices for the zr positions in the sequence where an edge contributing Yr occurs. This holds regardless of z2 , z3 , . . . , zr−1 . The second binomial coefficient bounds the number of choices for the dr −zr positions in the sequence where we choose an edge incident with ur after time tr . Conversely, the denominator in (5) is a lower bound on the number of choices for the dr positions where we choose an edge incident with ur , given d2 , d3 , . . . , dr−1 . We subtract the extra s to (over)count for edges from v to ur+1 , . . . , us . Expanding P(Zr = zr , r = 2, . . . , s | TS , DS , L) as a product of s = O(log n) of these terms completes the proof of Claim 4.2. 2
Going back to (4) we see that given d2 , d3 , . . . , ds , P (Ai2 ∧ · · · ∧ Ais | TS , DS , L) .
dr s X Y
P(Bin(zr , q
−1
) ≤ q − 1) ×
r=2 zr =0
≤
dr s X Y
1+
C2 zr + 1
C2 zr + 1
r=2 zr =0
≤
dr s X Y r=2 zr =0
1+
tr zr
m−tr dr −zr m dr
!
zr 1 1 zr 1− × min {zr , q − 1} q q−1 q m−tr ! tr dr −zr zrq−1 e−zr /q × zr m .
dr
7
tr zr
m−tr dr −zr m dr
! (6)
(7)
Here, C2 = C2 (q) depends only on q. We will use constants C3 , C4 , . . . in a similar fashion without further comment. Justification for (6): If zr ≤ q − 1 then P(Bin(zr , q −1 ) ≤ q − 1) = 1 and C2 = eq q will suffice. If q ≤ zr ≤ 10q we use P(Bin(zr , q
1 zr 1 zr 1 10q 1 1− ≥ q−1 1 − q − 1 q q−1 q q q
−1
) ≤ q − 1) ≤ 1 and
and C2 = e20 q q will suffice in this case.
zr 1 i qi
If zr > 10q then putting ai := P(Bin(zr , q −1 ) = i) =
1−
1 q
zr −i
for i ≤ q − 1 we see that
ai zr − i + 1 zr 5 1 zr − q = > 2 ≥ . · ≥ 2 ai−1 i q−1 q 2q q So here P(Bin(zr , q
−1
) ≤ q − 1) =
q−1 X
2 q−2 ! 2q 2q 2 + ··· + ≤ 1+ zr zr zr 1 1 zr 1 1−q 1− 1− q q − 1 q q−1 q
ai ≤ aq−1
i=0
q q−1 − 5 q 5 −1
and thus C2 = (5q)q suffices. Now, writing (t)z for the falling factorial t!/(t − z)! = t(t − 1)(t − 2) . . . (t − z + 1), m−tr tr dr (tr )zr (m − tr )dr −zr zr dr −zr = m zr (m)dr dr zY dr −z −1 Yr −1 m − tr − i dr r tr − i = · zr m − (dr − zr ) − i m−i i=0 i=0 dr −zr 2 zr dr tr tr dr . 1− ≤ 1+O m zr m m
1
,
(8)
Observe next that if zr ≥ q 2 then (zr )q−1 =
zrq−1
q−1 Y i=0
i 1− zr
≥
zrq−1
q2 1− 2zr
≥
zrq−1 . 2
(9)
It follows from (8) and (9) that dr X zr =q 2
≤ 2C2
C2 1+ zr + 1 dr X
zrq−1 e−zr /q ×
dr (zr )q−1 zr
zr =q−1
≤ 2C2 (dr )q−1
tr m
tr zr
tr e−1/q m
m−tr dr −zr m dr
!zr
tr 1− m
q−1 X dr dr − q + 1 zr − q + 1 zr =q−1
8
dr −zr
tr e−1/q m
!zr −q+1
tr 1− m
dr −zr
dr −q+1 tr −1/q = 2C2 1− 1−e m (dr − q + 1)tr dr tr q−1 −1/q exp − . 1−e ≤ 2C2 m m
dr tr m
q−1
Furthermore, not forgetting 2 −1 qX
zr =0
C2 1+ zr + 1
zrq−1 e−zr /q ×
tr zr
m−tr dr −zr m dr
≤ C3
2 −1 qX
tr zr
zr =1
≤ C4
2 −1 qX
tzrr ·
zr =1 2 −1 qX
m−tr dr −zr m dr
(m − tr )dr −zr dr ! · dr (dr − zr )! m
dr tr ≤ C4 m zr =1 dr tr , ≤ C5 ψ m
zr
e−(dr −zr )tr /m
P 2 −1 z 2 x . (Now zr ≤ q 2 and so the factor ezr tr /m ≤ eq can be absorbed into where ψ(x) = e−x qz=1 C5 .) Going back to (7) we have P (Ai2 ∧ · · · ∧ Ais | TS , DS , L) ≤ C6s
s Y r=2
dr tr m
q−1
! d t dr tr r r +ψ . (10) exp − 1 − e−1/q m m
It follows from (3) and (10) that, pv := P(|Cv(m) | > θv | TS , DS , L) X
≤
X
S⊂[2,k] ti :i∈[2,k]\S |S|=q−2−θv
s ti2 C6s Y m s
r=2
dr tr m
q−1
! dr tr d t r r exp − (1 − e−1/q ) + ψ . m m
Replacing a sum of products by a product of sums and dividing by (s−1)! to account for repetitions, we get !! s m X Y X dr t q−1 dr t C6s dr t −1/q pv ≤ exp − (1 − e ) +ψ m m m m s (s − 1)! S⊂[2,k] |S|=q−2−θv
r=3
t=1
×
m X t=1
t
d2 t m
q−1
!! d2 t d2 t −1/q exp − (1 − e ) + tψ . m m
We now replace the sums by integrals. This is valid seeing as the summands have a bounded number of extrema, and we replace C6 by C7 to absorb any small error factors. # ! Z ∞ " s X Y dr t q−1 dr t C7s dr t −1/q (1 − e ) +ψ pv ≤ exp − dt m m m m t=0 s (s − 1)! r=3 S⊂[2,k] |S|=q−2−θv
9
Z
∞
×
t t=0
=
X S⊂[2,k] |S|=q−2−θv
d2 t m
q−1
Z s Y C7s m ∞ q−1 −(1−e−1/q )x (x e + ψ(x))dx m dr x=0 s (s − 1)! r=3 m2 × 2 d2
≤
X S⊂[2,k] |S|=q−2−θv
≤
! ! d2 t d2 t −1/q exp − dt (1 − e ) +ψ m m
C7s · m s (s − 1)!
C8 m minr {dr }
Z
∞
n o −1/q x exp −(1 − e )x + xψ(x) dx q
x=0
s
C9k k q−2+θv . (log n)k−q+2+θv
Applying Lemmas 3.3 and 3.5 and removing the conditioning on TS , DS , L we see that with k0 = log n 100q , P(∃v : |Cv(m) | > θv ) ≤ P(¬L) +
20X log n k0 X e2ω (log n)k−q+1 C9k k q−2+θv C9k k q−2 × + n (k − 1)! (log n)k−q+2+θv (log n)k−q+2
k=q−1
≤ o(1) +
k=k0
20X log n e2ω X C9k k q−2+θv (20C9 )k +n log n (k − 1)! (log n)k/2 k≥q−1
≤ o(1) +
e2ω log n
X k≥q−1
k=k0
e)k
(C9 k k−q+2+θv
= o(1).
We show next that at time m, w.h.p. sets of size up to Ω(n) have large neighbourhoods in every color. We first prove that typically “large-degree vertices have large degree in every color”: let d∗c (v) denote the number of edges incident with v that COL colors c, except for those edges that are colored in Step 3. Theorem 4.3. There exists = (q) > 0 such that w.h.p. on completion of COL every v ∈ LARGE log n for all c ∈ [q]. has d∗c (v) ≥ 1000q log n Suppose we define a vertex to be smallc if it has dc (v) ≤ 1000q . Theorem 4.3 says w.h.p. the set of smallc vertices SM ALLc ⊂ SM ALL so that by Lemma 3.2, G does not contain any smallc structures of constant size. Here a smallc structure is a small structure made up of smallc vertices. The proof of Theorem 4.3 will follow from Lemmas 4.4, 4.5 and 4.6 below.
Lemma 4.4. There exists δ = δ(q) > 0 such that the following holds w.h.p.: Let F ull0 , F ull be as 1−δ . in Definition 2.1. Then |F ull0 | ≥ n − 203qn log n , and |F ull| ≥ n − n 10
Proof. We first note that for v ∈ [n], that if t = n log n then log n (t /2) P d (v) < λ0 := ≤ 3n−/2 < n−/3 . 100 Indeed, with p1 =
t /2
(n2 )
(11)
we see that, in the random graph model Gn,p1 :
λX 0 −1
nep1 λ0 −+o(1) n i n λ0 n−λ0 n−i ≤2 p (1 − p1 ) ≤2 p (1 − p1 ) P (d(v) < λ0 ) = n ≤ n−/2 . i 1 λ0 1 λ0 i=0 (12) Equation (11) now follows from (2) (with p replaced by p1 ) and (12). Thus the Markov inequality shows that with probability at least 1 − n−/3 , at least n − n1−/6 log n of the vertices v have d(t /2) (v) ≥ 100 . Now note that at most qn of the first t /2 edges were restricted in color by being incident to at least one needy vertex. This is because each time a needy vertex gets an edge incident to it, the total number of needed colors decreases by at least log n log n one. Therefore at most 200qn log n of these vertices v have fewer than 200 of their ≥ 100 initial edges colored completely at random, as in Step 4 of COL. Hence, there are at least n − d(t /2) (v)
log n 100
201qn log n
vertices v
log n 200
which have ≥ and edges of fully random color. For such a v, and any color c, log n log n 1 log n 1 16 log n (t /2) P dc (v) < ≤ P Bin ≤ ≤ exp − · ≤ n−/1000q . , · 1000q 200 q 1000q 2 25 200q (13) So P(v ∈ / F ull0 ) ≤ qn−/1000q , and the Markov inequality shows that w.h.p. |F ull0 | ≥ n −
201qn 202qn − n1−/2000q ≥ n − . log n log n
Now, for v ∈ / F ull0 , let S(v) := F ull0 \ N (t /2) (v). Since d(t /2) (v) ≤ d(v) ≤ 20 log n, we have |S(v)| ≥ n − 203qn log n . Furthermore, every w ∈ S(v) is no longer needy, and so among the next t /2 edges, at most q of the edges between v and S(v) have their choice of color restricted by v, and log n the rest are colored randomly as in Step 4 of COL. Now P(|e(v, S(v))| < 100 ) = O(n−/3 ) by a log n similar calculation to (12). Conditioning on |e(v, S(v))| ≥ 100 we have P(v ∈ / F ull) ≤ qn−/1000q 1−/2000q by a similar calculation to (13) and so |F ull| ≥ n − n with probability ≥ 1 − O(n−/2000q ) by the Markov inequality. We are working towards showing that vertices with low degree in some color must have have a low overall degree. The point is that all F ull vertices no longer need additional colors later than t = n log n, so any new edge connecting F ull to V \ F ull after time t has its color determined by log n the vertex not in F ull, as in Step 2 of COL. Indeed, suppose a vertex v ∈ / F ull has at least 400 log n log n edges to F ull after time t . Then v gets at least 400q > 1000q edges of every color incident with it. log n Lemma 4.5. W.h.p. there are no vertices v ∈ / F ull with at least 200 edges after time t i.e., log n log n (m) (t ) d (v) − d (v) ≥ 200 but with at most 400 of these edges to F ull. log n Proof. Take any vertex v ∈ / F ull and consider the first 200 edges incident to v after time t . We must estimate the probability that at least half of these edges are to vertices not in F ull. We bound this by log n/400 log n/200 n1−δ = o(1/n). log n/400 n − 20 log n
11
We subtract off a bound of 20 log n on the number of edges from v to F ull in Et . Note that we do not need to multiply by the number of choices for F ull, as F ull is defined by the first t edges. There at most n choices for v and so the lemma follows. Lemma 4.6. There are no large vertices v with d(m) (v) − d(t ) (v)
2|R| P ∃|R| ≤ (log n)3 n/(log n)3 r X n 2 ≤ p2r r 2r r=4 !2 r n/(log n)3 1+o(1) X log n ne re ≤ r 4n r=4
n/(log n)3
≤
X r=4
r e3+o(1) (log n)2 · n 16
!r = o(n−3 ).
Proof of Lemma 4.9: Case 1: |S| ≤ (lognn)4 . We may assume that S ∪ Nc∗ (S) is small enough for Claim 4.10 to apply (otherwise |Nc∗ (S)| ≥ n − (lognn)4 so that S actually has logarithmic expansion in color c). Then, using ec to denote (log n)3 the number of edges in color c, and using Theorem 4.3, X log n |S| ≤ d∗c (v) = 2ec (S) + ec (S, Nc∗ (S)) ≤ 4|S| + 2|Nc∗ (S) ∪ S|. 1000q v∈S
Hence, |Nc∗ (S)| ≥
log n |S| ≥ 19|S|, 2001q
which verifies the truth of the lemma for this case. Case 2: Let
n (log n)4
≤ |S| ≤
n 50 log n .
m+ :=
n log n . 8q
Let Ec+ , Ec∗ denote the edges of E + , E ∗ respectively, which are colored c. We begin by proving 13
Claim 4.11. |Ec+ |, |Ec∗ | ≥ m+ w.h.p. Proof. Once F ull has been formed, it follows from Lemma 4.4, that at most (n1−δ (n − n1−δ )) + n1−δ < 2n−δ n2 spaces remain in E(F ull, V \ F ull) or E(V \ F ull). For each of the m − t ∼ 2 ( 12 − )n log n edges appearing thereafter, since . n log n < n−δ n2 edges have been placed already, each has a probability ≥ 1−4n−δ of having both ends in F ull, independently of what has happened previously. Applying the Chernoff bounds (see for example [7], Chapter 21.4) we see that the probability that fewer than 13 n log n of these ( 12 − )n log n edges were between vertices in F ull is at most e−Ω(n log n) . We remind the reader that every edge with both endpoints in F ull is randomly colored and placed in E + or E ∗ in Step 3 of COL. 1 So, we may assume there are at least 3q n log n of these edges in E + ∪ E ∗ of color c in expectation 1 n log n = m+ w.h.p. in both E + and and then the Chernoff bounds imply that there are at least 8q ∗ E . Suppose there exists S as above with |Nc∗ (S)|
